We introduce the infinite random simplicial complex Δ, a simplicial complex on a countable set of vertices. The infinite random
simplicial complex Δ is the simplicial complex obtained almost surely by the following procedure. For each pair of vertices, draw an edge between
them with probability 1/2. Proceeding inductively, suppose that the (k-1)-skeleton Δk-1 of Δ has already been constructed.
Let v0,...,vk be a (k+1)-tuple of vertices of Δk-1 with the property that every proper subset of
{v0,...,vk} forms a face of Δk-1; we add the k-dimensional face
{v0,...,vk} to the k-skeleton Δk of Δ with probability 1/2.

We analyze in detail Manin's unirationality construction for del Pezzo surfaces of degree two with a point, extending his results and clarifying an
oversight. We also show that del Pezzo surfaces of degree two over a finite field are unirational with at most three possible exceptions:

We show that quartics over the complex numbers with at least eight hyperinflection lines are determined by their inflection lines. The result uses
the classification of quartics with at least eight hyperinflection lines by Vermeulen and exploits the fact that all these quartics have dihedral groups
in their automorphism groups. Our methods apply also in positive characteristic. Amusingly we
find that the three plane quartics over F13 with equations

We show that the K3 surface arising from Büchi's problem is the Kummer surface of the Jacobian J of the genus two curve branched over an arithmetic progression of length 5. We then exploit that the abelian surface
J is isogenous to a product of elliptic curves to conclude that the set of rational points on Büchi's K3 surface is Zariski dense. Finally we give a natural modular interpretation of Büchi's K3 surface as an
irreducible component of the moduli space of rank two stable vector bundles on the Büchi K3 surface itself, opening the way for a modular solution to Büchi's problem.

We start with the following question: given a number field L and a rational function F(x) with coefficients in L, when are there infinitely many values α in L such that F(α)
is a rational number? We generalize this question to the question of when does a fiber product of curves contain an irreducible component of genus at most one and we settle completely the problem when the curves in the fiber
product have genus one.

We start investigating to what extent plane curves can be reconstructed from the knwoledge of their inflection lines. For plane cubics, we prove that
the reconstruction is always possible. For general plane quartics, we show that the inflection lines and one inflection point are enough to
reconstruct the curve.

We study the Cox rings of rational surfaces for which the anti-canonical divisor is (essentially) effective, finding conditions under which the Cox
ring is finitely generated. We show that there is a curve of arithmetic genus one and a finitely generated subgroup of its Picard group that is finite if
and only if every nef divisor on the surface is semi-ample. Thus we are able reduce the question of finite generation of Cox rings to the question of
whether the semigroup of effective curves on the surface is finitely generated. We conclude with an example of a Mori dream rational surface with
vanishing anticanonical Iitaka dimension.

We compute the Picard group of the surface S of cuboids; the emphasis is to prove that a very explicit set of 140 curves on S generates the full
Picard group, not just a finite index subgroup. The main tool is the use of a combination of the geometric automorphism group of S, as well as the Galois
action on the set of curves to reduce the statement to the primitivity of the canonical class on S.

By a combination of the determinant method and n-descent on curves of genus one, we find bounds for the number of rational points on a smooth plane
cubic curve that are uniform in the coefficients and depend explicitly on the rank of the Mordell-Weil group of the curve. The argument exploits an
interplay between analytic techniques to bound the number of solutions to a cubic equation and a geometric interpretation of an evaluation map on an
abelian surface leading to the required inequalities.

We construct explicit models for the two-coverings of Jacobians of genus two curves and also for some of their twists. An important role is played by
the analysis of the action of the two-torsion subgroup on the equations of a natural embedding of the Jacobian of a curve of genus two in projective
space, as well as the induced action on the Kummer variety.

We present a unified approach to proving that smooth projective rational surfaces with big anti-canonical divisor have finitely generated Cox ring.
Almost every previously known example was treated by ad hoc arguments on a case-by-case basis and is covered by our general result; very few known
examples of rational surfaces for which this statement was previously known are not covered by our result. We also provide many new examples. An
explicit presentation in terms of generators and relations of the Cox ring of a variety is the main step in the explicit construction of a universal
torsor on the variety itself.

We settle a conjecture of Batyrev and Popov on the ideal of relations of the Cox ring of a del Pezzo surface in the last remaining case of del Pezzo
surfaces of degree one. Using techniques from commutative algebra we transform the problem to a geometric question about the vanishing of certain
cohomology groups of line bundles on the surface. We then analyze geometrically the various line bundles to reduce further the problem to a combinatorial
question on configurations of exceptional curves on the surface. We finally solve the combinatorial question by explicit geometric arguments.

A generalization of a result of Roya Beheshti and Jason Starr. The results in this paper imply that, for n>4, a smooth hypersurface of degree n in
n-dimensional projective space cannot be covered by Fano varieties (resp. toric varieties) of dimension between 2 and n-3 (inclusive). (For comparision,
Roya and Jason prove the same statement but only for Fano subvarieties of dimension 2, rather than the full range.) This is supporting evidence to the
widely believed conjecture that there exist rationally connected varieties that are not unirational and that, more precisely, general hypersurfaces of
degree n in P^n should be examples, for n large enough.

We introduce the notions of spherical and conical graphs, establishing links with (independent) dominating sets, edge covers and the homotopy type of
associated simplicial complexes. We also prove a formula to compute the Euler characteristic of a simplicial set.

A follow up to the previous paper. We generalize our reduction technique to cover a wider range of simplicial complexes. While the complexes that we
could analyze in the previous paper were either contractible or homotopy equivalent to spheres, the complexes here can also be (disjoint unions of) wedges
of spheres. This added flexibility and generality has as a drawback that there are several discrete invariants associated to a given complex and thus a
direct combinatorial characterization of the homotopy type is more difficult.

A combinatorial paper. We introduce a "reduction technique" to study the homotopy type of simplicial complexes, with the goal of answering a question
of Ehrenborg and Hetyei on the dimension of certain complexes known to be homotopy equivalent to spheres. As main applications we give combinatorial
interpretations of these dimensions in several widely studied classes of graphs associated to forests.

My first paper on Cox rings. Using commutative algebra arguments we exploit the action of the automorphism group of the Picard lattice of a del Pezzo
surface to reduce a question of Batyrev and Popov on the Cox rings of del Pezzo surfaces to a completely mechanical verification. The main application is
that we are able to use the computer to verify the conjecture for del Pezzo surfaces of degree at least three, a range that is still unreachable by brute
force computations and that was also not known at the time. Later generalizations and computer-free proofs were found (also by myself!).

The main result is the proof that the spaces of rational curves on del Pezzo surfaces representing a fixed divisor class are either empty or
irreducible, with one exception. The argument involves a detailed analysis of the possible singularities of (a partial resolution of) the moduli spaces
of rational curves on surfaces and combines this with a degeneration argument to reduce all the rational curves in a fixed divisor class to a standard
form. The relevant moduli spaces are sufficiently smooth that we can show that the various degenerations always take place within the smooth locus,
proving the required irreducibility. (The longer, actual PhD thesis is available here.)